Fix Me or Crush Me
Why do this problem?
This problem gives students the opportunity to explore the effect of matrix multiplication on vectors, and lays the foundations for studying the eigenvectors and kernel of a matrix, ideas which are very important in higher level algebra with applications in science.
Students might like to use this Matrix Multiplication calculator to test out their ideas.
Possible approach
Start by asking students to work with the vector ${\bf F}$ to find a matrix which fixes it. Initially, let students find their own methods of working - some may choose to try to fit numbers in the matrix, some may straight away work with algebra. Once students have had a chance to try the task, allow some time to discuss methods, as well as the simplest and most complicated examples of matrices they have managed to find.
Key questions
Very hard extension: Imagine that you are given a vector ${\bf F}$ and a vector ${\bf Z}$. Investigate whether you will be able to make a matrix $M$ which both fixes ${\bf F}$ and crushes ${\bf Z}$.
There are more matrix problems in this feature.
You may also like
Flexi Quads
A quadrilateral changes shape with the edge lengths constant. Show the scalar product of the diagonals is constant. If the diagonals are perpendicular in one position are they always perpendicular?
A Knight's Journey
This article looks at knight's moves on a chess board and introduces you to the idea of vectors and vector addition.
8 Methods for Three by One
This problem in geometry has been solved in no less than EIGHT ways by a pair of students. How would you solve it? How many of their solutions can you follow? How are they the same or different? Which do you like best?